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1、原理
若要使单摆不受影响水平方向的加速度影响而摆动，必须使单摆的 摆长等于地球半径
2、舒勒摆
摆长L等于地球半径 R 的单摆（不受加速度干扰的单摆）
3、舒勒周期
舒勒摆的固有振动周期 TTT 为84.484.484.4..." />
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1、原理
若要使单摆不受影响水平方向的加速度影响而摆动，必须使单摆的 摆长等于地球半径
2、舒勒摆
摆长L等于地球半径 R 的单摆（不受加速度干扰的单摆）
3、舒勒周期
舒勒摆的固有振动周期 TTT 为84.484.484.4...">
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1、原理
若要使单摆不受影响水平方向的加速度影响而摆动，必须使单摆的 摆长等于地球半径
2、舒勒摆
摆长L等于地球半径 R 的单摆（不受加速度干扰的单摆）
3、舒勒周期
舒勒摆的固有振动周期 TTT 为84.484.484.4...">
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                                <h2>
                                    2.5 舒勒原理
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                                    2023-10-03, 465 words, 2 min read
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                                        <h2 id="一-舒勒原理">一、舒勒原理</h2>
<h4 id="1-原理">1、原理</h4>
<p>若要使单摆不受影响水平方向的加速度影响而摆动，必须使单摆的 摆长等于地球半径</p>
<h4 id="2-舒勒摆">2、舒勒摆</h4>
<p>摆长L等于地球半径 R 的单摆（不受加速度干扰的单摆）</p>
<h4 id="3-舒勒周期">3、舒勒周期</h4>
<p>舒勒摆的固有振动周期 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span></span></span></span> 为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>84.4</mn></mrow><annotation encoding="application/x-tex">84.4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">8</span><span class="mord">4</span><span class="mord">.</span><span class="mord">4</span></span></span></span> 分</p>
<h4 id="4-意义">4、意义</h4>
<p>在近地惯性导航系统中，<strong>平台必须精确跟踪当地水平面</strong>（即平台竖轴必须精确跟踪当地垂线），以 便精确地给出加速度的测量基准。</p>
<p><strong>舒勒原理为测量基准提供理论条件，为制造高精度实用惯性导航系统提供了理论准备。</strong></p>
<h4 id="5-舒勒原理推导">5、舒勒原理推导</h4>
<img src="http://cos.pansis.site/202309281412042.png/abc123" alt="image-20230928141228738" style="zoom:33%;" />
<p>设运载体起始位置<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span></span></span></span>，加速度为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">AA</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span><span class="mord mathdefault">A</span></span></span></span>为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span></span></span></span>点地垂线；经过一段时间到任意位置<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>B</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">BB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span></span></span>为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span></span></span>点垂线。</p>
<p><strong>摆的受力分析：</strong></p>
<img src="http://cos.pansis.site/202309281415412.png/abc123" alt="image-20230928141549242" style="zoom:40%;" />
<p><strong>摆的力矩分析：</strong></p>
<img src="http://cos.pansis.site/202309281416499.png/abc123" alt="image-20230928141629315" style="zoom:40%;" />
<p><strong>运动方程式</strong></p>
<img src="http://cos.pansis.site/202309281436009.png/abc123" alt="image-20230928143645905" style="zoom: 50%;" />
<img src="http://cos.pansis.site/202309281437942.png/abc123" alt="image-20230928143708748" style="zoom:50%;" />
<h2 id="二-舒勒谐振">二、舒勒谐振</h2>
<h4 id="1-物理摆控制方块图">1、物理摆控制方块图</h4>
<p>1、相关运动方程式</p>
<p>当物理摆相对当地垂线的偏角<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span>为小角度时，</p>
<p>单摆运动方程式<img src="http://cos.pansis.site/202309281450642.png/abc123" alt="image-20230928145005567" style="zoom: 50%;" /></p>
<p>推导为 方程1 <img src="http://cos.pansis.site/202309281450953.png/abc123" alt="image-20230928145034911" style="zoom: 50%;" /></p>
<p>同时还有方程2和3  <img src="http://cos.pansis.site/202309281452574.png/abc123" alt="image-20230928145254496" style="zoom: 50%;" /></p>
<p><img src="http://cos.pansis.site/202309281453946.png/abc123" alt="image-20230928145322888" style="zoom:33%;" /><img src="http://cos.pansis.site/202309281453953.png/abc123" alt="image-20230928145350871" style="zoom:35%;" /></p>
<p>2、流程图</p>
<img src="http://cos.pansis.site/202309281454778.png/abc123" alt="image-20230928145445642" style="zoom:50%;" />
<p>3、要使摆线始终指向当地垂线，则 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>θ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\theta=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span> ，即 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>θ</mi><mi>b</mi></msub><mo>=</mo><msub><mi>θ</mi><mi>a</mi></msub></mrow><annotation encoding="application/x-tex">\theta_b=\theta_a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<p>则需要满足  <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>m</mi><mi>l</mi><mi mathvariant="normal">/</mi><mi>J</mi><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">ml/J=1/R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">m</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.09618em;">J</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span></span></p>
<h4 id="2-舒勒谐振的作用">2、舒勒谐振的作用</h4>
<ul>
<li>在近地惯性导航系统中，<strong>平台必须精确跟踪当地水平面</strong>（即平台竖轴必须精确跟踪当地垂线），以 便精确地给出加速度的测量基准</li>
<li>欲使平台精确跟踪当地水平面，就必须使平台 不受运载体加速度的干扰，因此<strong>平台的水平控制回路必须满足舒勒调谐条件</strong>。</li>
</ul>
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